We consider variants of the triangle-avoidance game first defined by Harary and rediscovered by Hajnal a few years later. A graph game consists of two players beginning with an empty graph on n vertices. The two players take turns choosing edges within Kn, building up a simple graph. The edges must be chosen according to a set of restrictions R. The winner is the last player to choose an edge that does not violate any of the restrictions in R. For fixed n and R, one of the players has a winning strategy. We look at games where R includes bounded degree, triangle-avoidance, and/or connectedness, and determine the winner for all n.